Alex Degtyarev

Real Enriques surfaces

Topology of involutions.- Integral lattices and quadratic forms.- Algebraic surfaces.- Real surfaces: the topological aspects.- Summary: Deformation Classes.- Topology of real enriques surfaces.- Moduli of real enriques surfaces.- Deformation types: the hyperbolic and parabolic cases.- Deformation types: the elliptic and parabolic cases.

TOPOLOGICAL PROPERTIES OF REAL ALGEBRAIC VARIETIES: DU COTE DE CHEZ ROKHLIN

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

On deformations of singular plane sextics

We study complex plane projective sextic curves with simple singularities up to equisingular deformations. It is shown that two such curves are deformation equivalent if and only if the corresponding pairs are diffeomorphic. A way to enumerate all deformation classes is outlined, and a few examples are considered, including classical Zariski pairs; in particular, promising candidates for homeomorphic but not diffeomorphic pairs are found.

ALEXANDER POLYNOMIAL OF A CURVE OF DEGREE SIX

The complete description of the Alexander polynomial of the complement of an irreducible sextic in is given. Some general results about Alexander polynomials of algebraic curves are also obtained.

On the Number of Components of a Complete Intersection of Real Quadrics

Our main results-5pc]Please check the text is ok? concern complete intersections of three real quadrics. We prove that the maximal number B20(N) of connected components that a regular complete intersection of three real quadrics in ℙ N may have differs at most by one from the maximal number of ovals of the submaximal depth \([(N - 1)/2]\)of a real plane projective curve of degree \(d = N + 1\). As a consequence, we obtain a lower bound \(\frac{1} {4}{N}^{2} + O(N)\)and an upper bound \(\frac{3} {8}{N}^{2} + O(N)\)for B20(N).

Oka's conjecture on irreducible plane sextics

We partially prove and partially disprove Okas conjecture on the fundamental group/Alexander polynomial of an irreducible plane sextic. Among other results, we enumerate all irreducible sextics with simple singularities admitting dihedral coverings and find examples of Alexander equivalent Zariski pairs of irreducible sextics

ZARISKI k-PLETS VIA DESSINS D'ENFANTS

We construct exponentially large collections of pairwise distinct equisingular defor- mation families of irreducible plane curves sharing the same sets of singularities. The funda- mental groups of all curves constructed are abelian.

Topological classification of real Enriques surfaces

We complete the classification of the topological types of real Enriques surfaces started by V. Nikulin. The resulting list contains 87 topological types.

On the moduli space of real Enriques surfaces

Abstract We introduce a new invariant, Pontryagin-Viro form, of real algebraic surfaces. We evaluate it for real Enriques surfaces with non-negative minimal Euler characteristic of the components of the real part and prove that, when combined with the known topological invariants, it distinguishes the deformation types of such surfaces.

Finiteness and quasi-simplicity for symmetric

We compare the smooth and deformation equivalence of actions of finite groups on K3-surfaces by holomorphic and anti-holomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show that, first, in general the homological type of an action does not even determine its topological type, and second, that K3-surfaces X and ¯ X with the same Klein action do not need to be equivariantly deformation equivalent even if the induced action on H 2,0 (X) is real, i.e., reduces to multiplication by ±1. 1.1. Questions. In this paper, we study equivariant deformations of complexK3- surfaces with symmetry groups, where by a symmetry we mean an either holomor- phic or anti-holomorphic transformation of the surface. Although the automor- phism group of a particular K3-surface may be infinite, we confine ourselves to finite group actions and address the following two questions (see 1.4-1.6 for precise definitions): finiteness: whether the number of actions, counted up to equivariant deforma-